11.1Database System Concepts - 6 th Edition Chapter 11: Indexing and Hashing Basic Concepts Ordered Indices B + -Tree Index Files B-Tree Index Files Static.

11.1Database System Concepts - 6th Edition

Chapter 11: Indexing and HashingChapter 11: Indexing and Hashing

Basic Concepts

Ordered Indices

B+-Tree Index Files

B-Tree Index Files

Static Hashing

Comparison of Ordered Indexing and Hashing

Index Definition in SQL

Multiple-key access

B-tree style

Grid file

#Bitmap


Basic ConceptsBasic Concepts

Indexing mechanisms used to speed up access to desired data.

E.g., author catalog in library

Search Key - attribute or set of attributes used to look up records in a file, like author, title, etc.

An index file consists of records (called index entries) of the form

Index files are typically much smaller than the original file

Two basic kinds of indices:

Ordered indices: search keys are stored in sorted order

Hash indices: search keys are distributed uniformly across “buckets” using a “hash function”.

search-key pointer


Index Evaluation MetricsIndex Evaluation Metrics

Access types supported efficiently. E.g.,

records with a specified value in the attribute

or records with an attribute value falling in a specified range of values.

Access time

Insertion time

Deletion time

Space overhead


Ordered IndicesOrdered Indices

In an ordered index, index entries are stored sorted on the search key value. E.g., author catalog in library.

Primary index: in a sequentially ordered file, the index whose search key specifies the sequential order of the file.

Also called clustering index

The search key of a primary index is usually but not necessarily the primary key.

Secondary index: an index whose search key specifies an order different from the sequential order of the file. Also called non-clustering index.

Index-sequential file: ordered sequential file with a primary index.


Dense Index FilesDense Index Files

Dense index — Index record appears for every search-key value in the file.

E.g. index on ID attribute of instructor relation ( 此例正好是 PK)


Dense Index Files (Cont.)Dense Index Files (Cont.)

Dense index on dept_name, with instructor file sorted on dept_name (此例並非 PK)

注意 : 指標指到第一筆


Sparse Index FilesSparse Index Files Sparse Index: contains index records for only some search-key values.

Applicable when records are sequentially ordered on search-key

只有 primary index 才適用 To locate a record with search-key value K we:

Find index record with largest search-key value < K

Search file sequentially starting at the record to which the index record points


Sparse Index Files (Cont.)Sparse Index Files (Cont.)

Compared to dense indices:

Less space and less maintenance overhead for insertions and deletions.

Generally slower than dense index for locating records.

Good tradeoff: sparse index with an index entry for every block in file, corresponding to least search-key value in the block.


Multilevel IndexMultilevel Index If primary index does not fit in memory, access becomes

expensive.

Solution: treat primary index kept on disk as a sequential file and construct a sparse index on it.

outer index – a sparse index of primary index

inner index – the primary index file

If even outer index is too large to fit in main memory, yet another level of index can be created, and so on.

Indices at all levels must be updated on insertion or deletion from the file.


Multilevel Index (Cont.)Multilevel Index (Cont.)


Index Update: DeletionIndex Update: Deletion

Single-level index entry deletion:

Dense indices – deletion of search-key is similar to file record deletion.

Sparse indices –

if an entry for the search key exists in the index, it is deleted by replacing the entry in the index with the next search-key value in the file (in search-key order).

If the next search-key value already has an index entry, the entry is deleted instead of being replaced.

If deleted record was the only record in the file with its particular search-key value, the search-key is deleted from the index also.


Index Update: InsertionIndex Update: Insertion

Single-level index insertion:

Dense indices – perform a lookup using the search-key value appearing in the record to be inserted; if the search-key value does not appear in the index, insert it.

Sparse indices – if index stores an entry for each block of the file, no change needs to be made to the index unless a new block is created.

If a new block is created, the first search-key value appearing in the new block is inserted into the index.

Multilevel insertion and deletion: algorithms are simple extensions of the single-level algorithms


Secondary IndicesSecondary Indices

Frequently, one wants to find all the records whose values in a certain field (which is not the search-key of the primary index) satisfy some condition.

Example 1: In the instructor relation stored sequentially by ID, we may want to find all instructors in a particular department

Example 2: as above, but where we want to find all instructors with a specified salary or with salary in a specified range of values

We can have a secondary index with an index record for each search-key value


Secondary Indices ExampleSecondary Indices Example

Index record points to a bucket that contains pointers to all the actual records with that particular search-key value.

Secondary indices have to be dense

Secondary index on salary field of instructor


Primary and Secondary IndicesPrimary and Secondary Indices

Indices offer substantial benefits when searching for records.

BUT: Updating indices imposes overhead on database modification --when a file is modified, every index on the file must be updated,

Sequential scan using primary index is efficient, but a sequential scan using a secondary index is expensive

Each record access may fetch a new block from disk

Block fetch requires about 5 to 10 milliseconds, versus about 100 nanoseconds for memory access


BB++-Tree Index Files-Tree Index Files

Disadvantage of sequential index files performance degrades as file grows, since many overflow

blocks get created. Periodic reorganization of entire file is required.

Advantage of B+-tree index files: automatically reorganizes itself with small, local, changes, in

the face of insertions and deletions. Reorganization of entire file is not required to maintain

performance. (Minor) disadvantage of B+-trees:

extra insertion and deletion overhead, space overhead. Advantages of B+-trees outweigh disadvantages

B+-trees are used extensively

B+-tree indices are an alternative to sequential index files.


Example of B+-Tree 以 attribute name 建 index

一個 attribute 所有的 value 建一顆樹

The non-leaf levels of the B+-tree form a hierarchy of sparse indices.


BB++-Tree Index Files (Cont.)-Tree Index Files (Cont.)

All paths from root to leaf are of the same length

Each node that is not a root or a leaf has between n/2 and n children.

A leaf node has between (n–1)/2 and n–1 values

Special cases:

If the root is not a leaf, it has at least 2 children.

If the root is a leaf (that is, there are no other nodes in the tree), it can have between 0 and (n–1) values.

A B+-tree is a rooted tree satisfying the following properties:


BB++-Tree Node Structure-Tree Node Structure

Typical node

Ki are the search-key values

Pi are pointers to children (for non-leaf nodes) or pointers to records or buckets of records (for leaf nodes).

The search-keys in a node are ordered

K1 < K2 < K3 < . . . < Kn–1

)


Leaf Nodes in BLeaf Nodes in B++-Trees-Trees

For i = 1, 2, . . ., n–1, pointer Pi ( 左邊的指標 ) points to a file record with search-key value Ki,

If Li, Lj are leaf nodes and i < j, Li’s search-key values are less than or equal to Lj’s search-key values

Pn points to next leaf node in search-key order

Properties of a leaf node:


Non-Leaf Nodes in BNon-Leaf Nodes in B++-Trees-Trees

Non leaf nodes form a multi-level sparse index on the leaf nodes. For a non-leaf node with n pointers:

All the search-keys in the subtree to which P1 points are less than K1

For 2 i n – 1, all the search-keys in the subtree to which Pi points have values greater than or equal to Ki–1 and less than Ki

All the search-keys in the subtree to which Pn points have values greater than or equal to Kn–1


Example of BExample of B++-tree-tree

Leaf nodes must have between 3 and 5 values ((n–1)/2 and n –1, with n = 6).

Non-leaf nodes other than root must have between 3 and 6 children ((n/2 and n with n =6).

Root must have at least 2 children.

B+-tree for instructor file (n = 6)


Observations about BObservations about B++-trees-trees

Since the inter-node connections are done by pointers, “logically” close blocks need not be “physically” close.

The B+-tree contains a relatively small number of levels

Level below root has at least 2* n/2 values

Next level has at least 2* n/2 * n/2 values

.. etc.

If there are K search-key values in the file, the tree height is no more than logn/2(K)

thus searches can be conducted efficiently.

Insertions and deletions to the main file can be handled efficiently, as the index can be restructured in logarithmic time (as we shall see).


Queries on BQueries on B++-Trees-Trees Function Find: Find record with search-key value V.

1. C=root

2. While C is not a leaf node {

1. Let i be least value s.t. V Ki.

2. If no such exists, set C = last non-null pointer in C

3. Else { if (V= Ki ) Set C = Pi +1 else set C = Pi}

}

3. Let i be the least value s.t. Ki = V

4. If there is such a value i, follow pointer Pi to the desired record.

5. Else no record with search-key value k exists.


Queries on BQueries on B+-+-Trees (Cont.)Trees (Cont.)

If there are K search-key values in the file, the height of the tree is no more than logn/2(K).

A node is generally the same size as a disk block, typically 4 kilobytes

and n is typically around 100 (40 bytes per index entry).

With 1 million search key values and n = 100

at most log50(1,000,000) = 4 nodes are accessed in a lookup.

Contrast this with a balanced binary tree with 1 million search key values — around 20 nodes are accessed in a lookup

above difference is significant since every node access may need a disk I/O, costing around 20 milliseconds


Updates on BUpdates on B++-Trees: Insertion-Trees: Insertion

1. Find the leaf node in which the search-key value would appear

2. If the search-key value is already present in the leaf node

1. Add record to the file

2. If necessary add a pointer to the bucket.

3. If the search-key value is not present, then

1. add the record to the main file (and create a bucket if necessary)

2. If there is room in the leaf node, insert (key-value, pointer) pair in the leaf node

3. Otherwise, split the node (along with the new (key-value, pointer) entry) as discussed in the next slide.


Updates on BUpdates on B++-Trees: Insertion (Cont.)-Trees: Insertion (Cont.)

Splitting a leaf node:

take the n (search-key value, pointer) pairs (including the one being inserted) in sorted order. Place the first n/2 in the original node, and the rest in a new node.

let the new node be p, and let k be the least key value in p. Insert (k,p) in the parent of the node being split.

If the parent is full, split it and propagate the split further up.

Splitting of nodes proceeds upwards till a node that is not full is found.

In the worst case the root node may be split increasing the height of the tree by 1.

Result of splitting node containing Brandt, Califieri and Crick on inserting AdamsNext step: insert entry with (Califieri,pointer-to-new-node) into parent


BB++-Tree Insertion-Tree Insertion

B+-Tree before and after insertion of “Adams”


BB++-Tree Insertion-Tree Insertion

B+-Tree before and after insertion of “Lamport”


BB++-Tree Insertion (cont)-Tree Insertion (cont)

B+-Tree before and after insertion of “Ken”


Splitting a non-leaf node: when inserting (k,p) into an already full internal node N

Copy N to an in-memory area M with space for n+1 pointers and n keys

Insert (k,p) into M

Copy P1,K1, …, K n/2-1,P n/2 from M back into node N

Copy Pn/2+1,K n/2+1,…,Kn,Pn+1 from M into newly allocated node N’

Insert (K n/2,N’) into parent N

Read pseudocode in book!

Crick

Insertion in BInsertion in B++-Trees (Cont.)-Trees (Cont.)

Adams Brandt Califieri Crick Adams Brandt

Califieri


PracticePractice

Construct a B+-tree for the following set of key values:

(2, 3, 5, 7, 11, 17, 19, 23, 29, 31)

Assume that the tree is initially empty, values are added in ascending order, and the number of pointers that fit in one node is “four”. (That is, the order is 4.)


Updates on BUpdates on B++-Trees: Deletion-Trees: Deletion

Find the record to be deleted, and remove it from the main file and from the bucket (if present)

Remove (search-key value, pointer) from the leaf node if there is no bucket or if the bucket has become empty

If the node has too few entries due to the removal, and the entries in the node and a sibling fit into a single node, then merge siblings:

Insert all the search-key values in the two nodes into a single node (the one on the left), and delete the other node.

Delete the pair (Ki–1, Pi), where Pi is the pointer to the deleted node, from its parent, recursively using the above procedure.


Updates on BUpdates on B++-Trees: Deletion-Trees: Deletion

Otherwise, if the node has too few entries due to the removal, but the entries in the node and a sibling do not fit into a single node, then redistribute pointers:

Redistribute the pointers between the node and a sibling such that both have more than the minimum number of entries.

Update the corresponding search-key value in the parent of the node.

The node deletions may cascade upwards till a node which has n/2 or more pointers is found.

If the root node has only one pointer after deletion, it is deleted and the sole child becomes the root.


Examples of BExamples of B++-Tree Deletion-Tree Deletion

Deleting “Srinivasan” causes merging of under-full leaves

Before and after deleting “Srinivasan”


Examples of BExamples of B++-Tree Deletion (Cont.)-Tree Deletion (Cont.)

Deletion of “Singh” and “Wu” from result of previous example

Leaf containing Singh and Wu became underfull, and borrowed a value Kim from its left sibling

Search-key value in the parent changes as a result


Example of BExample of B++-tree Deletion (Cont.)-tree Deletion (Cont.)

Before and after deletion of “Gold” from earlier example

Node with Gold and Katz became underfull, and was merged with its sibling Parent node becomes underfull, and is merged with its sibling

Value separating two nodes (at the parent) is pulled down when merging Root node then has only one child, and is deleted


B-Tree Index FilesB-Tree Index Files

Similar to B+-tree, but B-tree allows search-key values to appear only once; eliminates redundant storage of search keys.

Search keys in nonleaf nodes appear nowhere else in the B-tree; an additional pointer field for each search key in a nonleaf node must be included.

Generalized B-tree leaf node: Figure (a)

Nonleaf node – pointers Bi are the bucket or file record pointers: Figure (b)


B-Tree Index File ExampleB-Tree Index File Example

B-tree (above) and B+-tree (below) on same data


B-Tree Index Files (Cont.)B-Tree Index Files (Cont.)

Advantages of B-Tree indices:

May use less tree nodes than a corresponding B+-Tree.

Sometimes possible to find search-key value before reaching leaf node.

Disadvantages of B-Tree indices:

Only small fraction of all search-key values are found early

Non-leaf nodes are larger, so fan-out is reduced. Thus, B-Trees typically have greater depth than corresponding B+-Tree

Insertion and deletion more complicated than in B+-Trees

Implementation is harder than B+-Trees.

Typically, advantages of B-Trees do not out weigh disadvantages.


Static HashingStatic Hashing

A bucket is a unit of storage containing one or more records (a bucket is typically a disk block).

In a hash file organization we obtain the bucket of a record directly from its search-key value using a hash function.

Hash function h is a function from the set of all search-key values K to the set of all bucket addresses B.

Hash function is used to locate records for access, insertion as well as deletion.

Records with different search-key values may be mapped to the same bucket; thus entire bucket has to be searched sequentially to locate a record.


Example of Hash File OrganizationExample of Hash File Organization

There are 10 buckets,

The binary representation of the ith character is assumed to be the integer i.

The hash function returns the sum of the binary representations of the characters modulo 10

E.g. h(Music) = 1 h(History) = 2 h(Physics) = 3 h(Elec. Eng.) = 3

Hash file organization of instructor file, using dept_name as key (See figure in next slide.)


Example of Hash File Organization Example of Hash File Organization

Hash file organization of instructor file, using dept_name as key (see previous slide for details).


Hash FunctionsHash Functions

Worst hash function maps all search-key values to the same bucket; this makes access time proportional to the number of search-key values in the file.

An ideal hash function is uniform, i.e., each bucket is assigned the same number of search-key values from the set of all possible values.

Ideal hash function is random, so each bucket will have the same number of records assigned to it irrespective of the actual distribution of search-key values in the file.

Typical hash functions perform computation on the internal binary representation of the search-key.

For example, for a string search-key, the binary representations of all the characters in the string could be added and the sum modulo the number of buckets could be returned. .


Handling of Bucket OverflowsHandling of Bucket Overflows

Bucket overflow can occur because of

Insufficient buckets

Skew in distribution of records. This can occur due to two reasons:

multiple records have same search-key value

chosen hash function produces non-uniform distribution of key values

Although the probability of bucket overflow can be reduced, it cannot be eliminated; it is handled by using overflow buckets.


Handling of Bucket Overflows (Cont.)Handling of Bucket Overflows (Cont.)

Overflow chaining – the overflow buckets of a given bucket are chained together in a linked list.

Above scheme is called closed hashing.

An alternative, called open hashing, which does not use overflow buckets, is not suitable for database applications.


Hash IndicesHash Indices

Hashing can be used not only for file organization, but also for index-structure creation.

A hash index organizes the search keys, with their associated record pointers, into a hash file structure.

Strictly speaking, hash indices are always secondary indices

if the file itself is organized using hashing, a separate primary hash index on it using the same search-key is unnecessary.

However, we use the term hash index to refer to both secondary index structures and hash organized files.


Example of Hash IndexExample of Hash Index

hash index on instructor, on attribute ID


Deficiencies of Static HashingDeficiencies of Static Hashing

In static hashing, function h maps search-key values to a fixed set of B of bucket addresses. Databases grow or shrink with time.

If initial number of buckets is too small, and file grows, performance will degrade due to too much overflows.

If space is allocated for anticipated growth, a significant amount of space will be wasted initially (and buckets will be underfull).

If database shrinks, again space will be wasted.

One solution: periodic re-organization of the file with a new hash function

Expensive, disrupts normal operations

Better solution: allow the number of buckets to be modified dynamically.

Dynamic hashing, extendable hashing, etc. have been proposed. We omit the discussion.


Comparison of Ordered Indexing and HashingComparison of Ordered Indexing and Hashing

Cost of periodic re-organization

Relative frequency of insertions and deletions

Is it desirable to optimize average access time at the expense of worst-case access time?

Expected type of queries:

Hashing is generally better at retrieving records having a specified value of the key.

If range queries are common, ordered indices are to be preferred

In practice:

PostgreSQL supports hash indices, but discourages use due to poor performance

Oracle supports static hash organization, but not hash indices

SQLServer supports only B+-trees


Index Definition in SQLIndex Definition in SQL

Create an index

create index <index-name> on <relation-name>(<attribute-list>)

E.g.: create index b-index on branch(branch_name)

Use create unique index to indirectly specify and enforce the condition that the search key is a candidate key.

Not really required if SQL unique integrity constraint is supported

To drop an index

drop index <index-name>

Most database systems allow specification of type of index, and clustering.


Multiple-Key AccessMultiple-Key Access

Use multiple (B-tree) indices for certain types of queries. Example:

select ID

from instructor

where dept_name = “Finance” and salary = 80000 Possible strategies for processing query using indices on single

attributes:

1. Use index on dept_name to find instructors with department name Finance; test salary = 80000

2. Use index on salary to find instructors with a salary of $80000; test dept_name = “Finance”.

3. Use dept_name index to find pointers to all records pertaining to the “Finance” department. Similarly use index on salary. Take intersection of both sets of pointers obtained.


Indices on Multiple KeysIndices on Multiple Keys

Composite search keys are search keys containing more than one attribute

E.g. (dept_name, salary)

Lexicographic ordering: (a1, a2) < (b1, b2) if either

a1 < b1, or

a1=b1 and a2 < b2


Indices on Multiple AttributesIndices on Multiple Attributes

With the where clause where dept_name = “Finance” and salary = 80000the index on (dept_name, salary) can be used to fetch only records that satisfy both conditions.

Using separate indices in less efficient — we may fetch many records (or pointers) that satisfy only one of the conditions.

Can also efficiently handle where dept_name = “Finance” and salary < 80000

But cannot efficiently handle where dept_name < “Finance” and balance = 80000

May fetch many records that satisfy the first but not the second condition

Suppose we have an index on combined search-key(dept_name, salary).


Grid FilesGrid Files

Structure used to speed the processing of general multiple search-key queries involving one or more comparison operators.

The grid file has a single grid array and one linear scale for each search-key attribute. The grid array has the number of dimensions equal to the number of search-key attributes.

Multiple cells of grid array can point to same bucket

To find the bucket for a search-key value, locate the row and column of its cell using the linear scales and follow pointer


Example Grid File for Example Grid File for accountaccount


Queries on a Grid FileQueries on a Grid File

A grid file on two attributes A and B can handle queries of all following forms with reasonable efficiency

(a1 A a2)

(b1 B b2)

(a1 A a2 b1 B b2),.

E.g., to answer (a1 A a2 b1 B b2), use linear scales to find corresponding candidate grid array cells, and look up all the buckets pointed from those cells.


Grid Files (Cont.)Grid Files (Cont.)

During insertion, if a bucket becomes full,

new bucket can be created if more than one cell points to it.

If only one cell points to it, either an overflow bucket must be created or the grid size must be increased

Linear scales must be chosen to uniformly distribute records across cells.

Otherwise there will be too many overflow buckets.

Periodic re-organization to increase grid size will help.

But reorganization can be very expensive.

Space overhead of grid array can be high.

R-trees (Chapter 23) are an alternative


#Bitmap Indices#Bitmap Indices

Bitmap indices are a special type of index designed for efficient querying on multiple keys

Records in a relation are assumed to be numbered sequentially from, say, 0

Given a number n it must be easy to retrieve record n

Particularly easy if records are of fixed size

Applicable on attributes that take on a relatively small number of distinct values

E.g. gender, country, state, …

E.g. income-level (income broken up into a small number of levels such as 0-9999, 10000-19999, 20000-50000, 50000- infinity)

A bitmap is simply an array of bits


#Bitmap Indices (Cont.)#Bitmap Indices (Cont.)

In its simplest form a bitmap index on an attribute has a bitmap for each value of the attribute

Bitmap has as many bits as records

In a bitmap for value v, the bit for a record is 1 if the record has the value v for the attribute, and is 0 otherwise



Bitmap indices are useful for queries on multiple attributes

not particularly useful for single attribute queries

Queries are answered using bitmap operations

Intersection (and)

Union (or)

Complementation (not)

Each operation takes two bitmaps of the same size and applies the operation on corresponding bits to get the result bitmap

E.g. 100110 AND 110011 = 100010

100110 OR 110011 = 110111 NOT 100110 = 011001

Males with income level L1: 10010 AND 10100 = 10000

Can then retrieve required tuples.

Counting number of matching tuples is even faster



Bitmap indices generally very small compared with relation size

E.g. if record is 100 bytes, space for a single bitmap is 1/800 of space used by relation.

If number of distinct attribute values is 8, bitmap is only 1% of relation size

Deletion needs to be handled properly

Existence bitmap to note if there is a valid record at a record location

Needed for complementation

not(A=v): (NOT bitmap-A-v) AND ExistenceBitmap

Should keep bitmaps for all values, even null value

To correctly handle SQL null semantics for NOT(A=v):

intersect above result with (NOT bitmap-A-Null)

11.1Database System Concepts - 6 th Edition Chapter 11: Indexing and Hashing Basic Concepts Ordered Indices B + -Tree Index Files B-Tree Index Files Static.

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